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Math 445 Assignment 7

Reading:

In class we are introducing projective geometry. In class we are following more the treatment of Ogilvy (from handouts) rather than Bix,but Chapter III of Bix has many details that may be helpful.

Bix, Ch3, Introduction, pp. 263-266

This introduction gives an orientation and some history.

Section 14, The Extended Plane, pp 266-282

This explains how points are added to the Euclidean plane to form the extended (projective) plane. Also, Pappus's Theorem is introduced in this section.

Section 15, Pappus' Theorem and Projections Between Planes.

Projections between planes are defined and it is shown how any line can be projected to the line at infinity. This is used to prove Pappus' Theorem.

Section 16, Desargues' Theorem and Duality.

The statement and proof of Desargues' Theorem are given. The concept of duality is introduced.

Section 17. Harmonic Sets.

Bix takes as definition the projective definition using a complete quadrangle. In the 445 course we have followed Ogilvy and defined harmonic sets by ratios and then see the complete quadrangle connection to harmonic sets as a theorem.

Part A. Due Wednesday, 2/24 -via email

Continue the reference exploration assignment of the previous week (when you found and reported on a book and an article) and file the same kind of email report about a resource on the Web. Again give a precise reference and email a description as last time.

Part B. Due Friday, 2/26 - on paper

7.1 Print out the sketch from Ex. 3, Equilateral Triangle from the stereographic sphere lab, Wed., 2/19.

7.2 Explore the behavior of the perpendicular bisectors of a spherical triangle. Produce one good figure, either with Sketchpad or on a spherical model. Write up your results, with good reasons or proofs. (Note: The three sides extended of a spherical triangle define 8 triangles on the sphere. How are the perpendicular bisectors related?)

7.3 Explore the behavior of the angle bisectors of a spherical triangle. Produce one good figure, either with Sketchpad or on a spherical model. Write up your results, with good reasons or proofs. (Note: The three sides extended of a spherical triangle define 8 triangles on the sphere. How are the angle bisectors related?)

Part C (due Friday 2/26) - by email

Write two or three paragraphs describing a possible topic for your project and how you would approach it in the project. (You can still change topics after this, but this should be a plausible scenario anyway.)

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